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Choosing between persistent and stationaryvolatility

I. Chronopoulo, L. Giraitis, G. Kapetanios

Working paper No. 2021/1 | February 2021

Choosing between persistent and stationary volatility

Ilias Chronopoulos∗† Liudas Giraitis ‡ George Kapetanios §

February 14, 2021

Abstract

In this paper, we analyse persistent and possibly non-stationary processes that have

the potential to characterise volatility better than stationary alternatives. We discuss

in detail both the conditions needed for their consistent estimation and conditions that

enable the use of standard ARCH tests to detect presence of stationary volatility af-

ter persistent volatility is taken into account. We provide Monte Carlo evidence that

supports our testing strategy in small samples and present extensive empirical evi-

dence clearly supporting the persistent volatility paradigm, suggesting that stationary

time–varying conditional volatility is less pronounced than previously thought. Fi-

nally, results from an out-of-sample forecasting exercise are presented, that support

our proposed persistent volatility paradigm.

Key words: Time-varying coefficient models, random coefficient models, non-

parametric estimation, kernel estimation, persistence, volatility.

JEL classification: C10, C12, C14.

∗Chronopoulos’s research is supported by the ESRC grant ES/P000703/1†King’s Business School, King’s College London, Email: Ilias.Chronopoulos@kcl.ac.uk‡ Queen Mary, University of London, Email: L.Giraitis@qmul.ac.uk§ King’s Business School, King’s College London, Email: George.Kapetanios@kcl.ac.uk

1

1 Introduction

Two important issues widely discussed in empirical econometric analysis for macroeconomics

and finance, over the last 25 years, are structural change and volatility modelling. Starting

with the seminal work of Engle (1982), volatility modelling has developed into a large topic of

study - perhaps the major preoccupation of financial econometrics. Most work has produced

models that are stationary but, crucially, allow for time variation in conditional variances.

There are two important groups of parametric models used to model volatility. Firstly,

the generalised ARCH class, where specific models contain a single innovation process and,

secondly, Stochastic Volatility (SV ) models, where the conditional variance is treated as a

latent variable and more than one innovation processes enter the model.

Empirical work though, has repeatedly concluded that variation in volatility can be

extremely persistent. Such a finding is not easily accommodated within the above stationary

model classes. The challenge is revealed in observed parameter estimates that are close to

the boundary of stationarity. This integrated GARCH effect, see e.g. Mikosch and Starica

(2004), can be caused by structural change in the unconditional variance, where it changes

either smoothly or abruptly over time. So it is possible that once allowed for, volatility can

be best characterised by persistent, and possibly nonstationary processes. There is a growing

literature that tries to characterise volatility using processes that allow for gradual change in

the unconditional variance. First, we succinctly summarise the main ways this is addressed

in the literature, and then present our main contributions.

The first line of research, following recent work on structural change, has focused on

paradigms coming from the statistical literature, such as the work of Priestley (1965) and

Dahlhaus (2000), where processes are smooth deterministic functions of time. Dahlhaus,

Rao, et al. (2006) proposed the locally stationary time-varying ARCH model, where sta-

tionarity is assumed locally, but the process is globally nonstationary. Along the same lines

Van Bellegem and Von Sachs (2004) proposed another smooth deterministic model, where

they assumed that volatility is multiplicatively decomposed into a stationary and nonsta-

tionary part. The assumption that there is a nonstationary part that could at least partially

drive the volatility, has recently permeated in standard ARCH (stationary) models as well.

Specifically, Engle and Rangel (2008), used exponential splines to specify the nonstationary

part. Brownlees and Gallo (2009) followed closely using a different version of splines. Mazur

and Pipien (2012), used the Fourier Flexible form of Gallant (1984), and finally, in a series

of papers Amado and Terasvirta (2013) and Amado and Terasvirta (2017), suggested the

use of a linear combination of logistic functions and their generalisations.

While the above characterisations provide an avenue to describe and estimate non-

stationary processes for the volatility process, either with kernel type methods or semi-

parametrically, there is no clear way to separate the two kinds. Further, the above char-

acterisations are tied to either parametric forms for the nonstationary component or are

subject to be smooth deterministic functions of time, which disallows the use of stochastic

2

components.

It is in this context that this paper makes a number of contributions. First, we discuss

possible setups for processes that are persistent and can potentially account for the persis-

tence of volatility observed in data. Then, we use ideas from recent work in the structural

change literature to discuss how persistence, perhaps surprisingly, allows estimation of the

unobserved volatility process without strong parametric assumptions and the requirement

that they are smooth and deterministic functions of time1. While such a focus on smooth

deterministic functions of time allows estimation using kernel methods, it is not satisfactory

as it does not allow for stochastic elements that can provide a richer representation.

Recent work by Giraitis, Kapetanios, and Yates (2014) address this by showing that as

long as a process satisfies a smoothness and boundedness (or moment) condition, then it

can be stochastic but still estimable using kernel estimation. This opens up the possibility

that such processes may adequately fit the observed behaviour of time-varying volatility,

being clearly more persistent than stationary processes. In fact this persistence is their most

distinctive characteristic. To appreciate this, it is important to focus on the estimation

strategy considered by Giraitis, Kapetanios, and Yates (2018). They essentially ask the

following question: Assuming a decomposition of the form yt = htut, for some observed

process yt, what properties should ht have, so that h2t can be consistently estimated by,

essentially, a rolling window type mean estimation of y2t , with an expanding window size?

It follows that the answer cannot lie within the class of stationary processes whose squares

satisfy a law of large numbers. As a result, the vast majority of stationary models do not

qualify. Instead, the answer seems to be that a process has to change slowly, in the sense that

|ht−hs| has to be small. Further, and for obvious reasons, it has to be bounded, a condition

that disallows the use of random walks, which would otherwise be of great relevance due to

their persistence. However, a normalised, and therefore bounded, random walk provides a

canonical example for the sort of processes we have in mind.

The ability to consistently estimate ht if it is persistent but not otherwise, provides a

clear avenue for a strategy to separate stationary from persistent processes, that is clearly

missing from the literature. If the unobserved volatility of a process yt is persistent then it

can be estimated and then the rescaled series yt/ht can be tested for the presence of time

varying stationary volatility using existing standard ARCH tests. If there is only persistent

volatility, the tests will not reject. If there is only stationary volatility, normalisation by

the estimate of h2t will not remove it, since the estimate will simply converge to E(h2t ), and

the tests will reject. Finally, the possibility exists that both the persistent and stationary

volatility components co-exist. Estimation of the persistent part can proceed in the presence

of the stationary part and then again the test will reject. In a second step, a stationary

volatility model can be fitted and estimated. This case is not covered in this paper and we

leave it for future work.

1see e.g. Priestley (1965), Dahlhaus (2000), Kapetanios and Yates (2008) and Van Bellegem and

Von Sachs (2004), among others.

3

In this paper, we discuss in detail conditions needed for consistent estimation of persistent

volatility scaling factor ht and further, conditions that enable the use of standard ARCH

tests to separate persistent volatility from stationary volatility. We provide illustrative Monte

Carlo results that support our approach in small samples. We proceed and present extensive

empirical evidence clearly supporting the persistent volatility paradigm, suggesting that

stationary time varying volatility is less pronounced than previously thought and that further

conditional second moments of asset returns change slowly. Finally, results from an out-of-

sample recursive forecasting exercise are presented and found to support the use of persistent

volatility modelling

The remainder of this paper is organised as follows. Section 2 presents our econometric

procedure and theoretical results. Section 3 contains the Monte Carlo exercise. Section 4

presents the empirical results from implementing our testing strategy to the data, and from

the forecasting exercise. Section 5 concludes. Proofs are relegated to the appendix.

2 Theoretical considerations

We consider the following white noise model:2

yt = htut, t = 1, . . . , T, (2.1)

where ut is a stationary sequence of uncorrelated random variables with Eut = 0, Eu2t = 1,

and ht is a persistent scale factor (stochastic or deterministic). We assume that sequences

{ut} and {ht} are mutually independent. Then

cov(yt, ys) = E[hths]E[utus] = 0 for t 6= s.

Given observations y1, ..., yT , our objective is to test for the presence of conditional het-

eroscedasticity (ARCH effect) in u2t , i.e. to determine whether ut is an i.i.d noise or depen-

dent random variables which in addition to ht contributes in (2.1) a stationary conditional

e.g. GARCH type volatility E[u2t |ut−1, ut−2, ...]. Since ut is not observed, we will estimate

ht by an estimate ht and base testing for ARCH effects on residuals ut = h−1t yt. Such test-

ing requires uniform consistency in estimation of u2t by u2t and thus, stronger conditions on

(ht, ut) than in point estimation of ht at time t. This is reflected in Assumptions M and H

we make on ut and ht, in particular the assumption of mutual independence of {h2t} and

{u2t} we impose. The latter clearly holds for a deterministic scaling factor ht.

2This model abstracts from the general case of a model with a specified conditional mean and time

varying volatility of the form yt = E (yt|Ft−1) + htut, t = 1, . . . , T, where Ft−1 is a sigma field: Ft−1 =

σ {ht−1, . . . , h0, ut−1, . . . , u1}, by setting E (yt|Ft−1) to zero for simplicity. The general case can be handled

straightforwardly.

4

Assumption M (α-mixing)

1. {ut} is a stationary ergodic sequence with Eut = 0, Eu2t = 1, Eutus = 0 for t 6= s, and

E|u1|θ <∞ for some θ > 6.

2. {ut} is α-mixing with mixing coefficients αk ≤ cφk, k ≥ 1, for some 0 < φ < 1 and c > 0.

Assumption H (Smoothness)

1. Variables h1, ..., hT satisfy the following smoothness condition. For some γ ∈ (1/2, 1],

|ht − hj| ≤ (|t− j|/T )γξtj, t, j = 1, ...., T (2.2)

and for some 0 < α ≤ ∞, c > 0

maxt,j≥1

E[exp(c|ξtj|α] <∞, maxt,j≥1

E[exp(c|ht|α)] <∞. (2.3)

2. There exists a > 0 such that for all t ≥ 1, ht ≥ a > 0 a.s.

3. Variables {ht} and {ut} are mutually independent.

Remark 2.1. 1. Condition (2.2) in Assumption H implies that the volatility process htdrifts slowly in time, which essentially rules out explosive behaviour. This is a widely used

assumption in the literature. It allows the use of both deterministic varying process of the

form ht = g(t/T ), where g(·) is a Lipschitz smooth function with parameter 1/2 < γ ≤ 1,

i.e. |g(x)− g(y)| ≤ C|x− y|γ, as well as a stochastic process ht.

2. The deterministic specification ht = g(t/T ), t = 1, ..., T is a standard assumption in

the work of Dahlhaus on locally stationary processes (see, e.g. Dahlhaus (2000) or Dahlhaus

and Polonik (2006)). The stochastic time-variation of ht was proposed by Giraitis et al.

(2014, 2016, 2018), to allow for stochastic processes that can be presented as non-stationary

random walks. The combinations between the two, satisfying (2.2) can be summarised as

ht = |T−γ(v1 + ...+ vt) + g(t/T )|+ a, t = 1, ..., T, (2.4)

where a > 0 and vt is a stationary zero mean sequence, see Example 2.1.

3. Our testing procedures will still work for the case of yt with a non-zero conditional

mean; in this case a first step estimator for the mean will be required, see e.g. Chronopoulos,

Kapetanios, and Petrova (2019).

Example 2.1. Let vj be a stationary zero mean sequence of Gaussian ARFIMA(p, d, q)

variables with parameter d ∈ (0, 1/2)3. Set γ = 1/2 + d. If |g(x) − g(y)| ≤ C|x − y|ν for

x, y ∈ (0, 1) where γ ≤ ν ≤ 1 then h2t in (2.4) satisfies Assumption H with γ = 1/2 + d and

α = 2. Indeed, for t > s,

|ht − hs| =∣∣T−γ∑t

j=1 vj + g(t/T )| −∣∣T−γ∑s

j=1 vj + g(s/T )∣∣

≤ |T−γ∑t

j=s+1 vj|+ |g(t/T )− g(s/T )| ≤ (|t− s|/T )γ|ξts|+ C(|t− s|/T )γ

3see Chapter 7 in Giraitis, Koul, and Surgailis (2012)

5

where ξtj = (t − s)−γ∑t

j=s+1 vj is a Gaussian r.v. Since var(ξtj) = var(ξt−j,0) → v2d as

t − j → ∞4, ξtj satisfies (2.2) with γ = 1/2 + d and α = 2. Recall that∑t

j=1 vj is

ARFIMA(p, 1 + d, q) process.

2.1 Volatility estimation

For the estimation of ht, we follow Giraitis, Kapetanios, and Yates (2018). They considered

estimation of ht in the context of VAR(1) model. We show that in the model described in

(2.1) and under Assumptions H and M, h2t can be consistently estimated as

h2t = K−1t

T∑j=1

b|t−j|y2j , Kt =

T∑j=1

b|t−j|, t = 1, ..., T, (2.5)

where b|t−j| = K((t − j)/H) are kernel weights. K(·) is assumed to be a non-negative and

bounded function, with piecewise bounded derivative, and H is a bandwidth parameter that

satisfies H = o(T ), as T →∞. Commonly used examples of K(x) include:

K(x) = (1/2)I(|x| ≤ 1), flat kernel,

K(x) = (3/4)(1− x2)I(|x| ≤ 1), Epanechnikov kernel,

K(x) = (1/√

2π)e−x2/2, Gaussian kernel.

The first two kernel functions have finite support, whereas the Gaussian kernel has infinite

support. We further assume that on its support,

K(x) ≤ C(1 + xν)−1, |(d/dx)K(x)| ≤ C(1 + xν)−1, x ≥ 0 for some ν > 3, C > 0. (2.6)

Under this setup, in Lemma 6.5 we show that

|h2t − h2t | = Op

((H/T )γ +H−1/2

), (2.7)

and in Lemma 6.3 using Dendramis, Giraitis, and Kapetanios (2021) results we show uniform

convergence

maxt=1,...,T

|h2t − h2t | = oP (1). (2.8)

The uniform convergence result on the asymptotic consistency in the estimation of ht, at a

non-parametric rate, will prove useful in our testing procedure for distinguishing between

persistent and stationary volatility that follows.

4see Proposition 3.3.1 in Giraitis, Koul, and Surgailis (2012)

6

2.2 Testing

In this subsection we consider how our strategy for discriminating between persistent and

stationary volatility works. First, let us briefly summarise the most basic tests used in

testing for the presence of ARCH effects for a stationary sequence of uncorrelated variables

yt = htut with ht = const.

To test for ARCH effect in u2t , we use the test Lagrange Multiplier (LM) test by Engle

(1982) which is similar to the LM test for autocorrelation. We fit to u2t an AR(p), p ≥ 1

model

u2t = β0 + β1u2t−1 + ...+ βpu

2t−p + ηt (2.9)

where β0 > 0 and test the following null hypothesis:

H0 : β1 = β2 = . . . = βp = 0

against the alternative

H1 : βj 6= 0 for some j = 1, ..., p.

Engle (1982) derived a Lagrange Multiplier (LM) statistic for testing H0, based on TR2,

where T is the sample size and R-squared is obtained from the auxiliary regression of u2t on

a constant and u2t−1, . . . , u2t−p. Under H0 (ut ∼ i.i.d.), the LM statistic follows asymptotically

a χ2p distribution. Further tests, such as the Wald and Likelihood ratio, have been shown to

be asymptotically equivalent to the LM test. Through testing, the literature mainly tries to

address two distinct problems: i) the misspecification of the conditional mean, see e.g. the

discussion in Bera and Higgins (1993) and ii) the correct specification of the volatility process.

Our work naturally falls in the second class, and in essence testing is useful to address

whether persistent processes, defined as the ones that follow the smoothness condition (2.2)

in Assumption H, provide a better specification for the volatility process.

Consider the model for yt in (2.1) where ut’s are not observed, and let the standardised

residuals be

ut =yt

ht, t = 1, . . . , T, (2.10)

where ht is defined as

√h2t of the kernel estimate of the persistent volatility using the

estimator described in (2.5). Our aim is to show that asymptotically, it is equivalent to

test for ARCH effects using the standardised residuals, u = [u1, u2, . . . , uT ]′, instead of u =

[u1, u2, . . . , uT ]′. Such an equivalence implies that if a process ht, that follows Assumption

H, drives the conditional second moment, then, under consistent estimation of ht using the

estimator ht in (2.5), the standardised residuals, ut = h−1t ut, should behave as white noise.

In Theorem 2.4 we show that ut can be used to compute the correlogram of ut and test for

absence of correlation in ut.

Testing for ARCH effects using regression (2.9) for squares u2t , note, that if stationary

processes (co-)drive the volatility via ut, then the normalisation by ht will not corrupt the

properties of testing. In our setup and for p ≥ 1, consider TS (u) = TR2, the test statistic

based on u and TS (u) = TR2 based on the standardised residuals, u, as described above.

7

The formulas of S (u) and S (u) are given in (6.1) and (6.2) of the proof. The following

Theorem, that is proven in the Appendix, gives the sufficient condition for LM test for ARCH

effects in squares u2t to be asymptotically valid when applied to u, instead of u.

Denote γk ≡ γu2,k = cov(u2k, u20), k ≥ 0 and set Γp =

(γ|j−k|

)j,k=1,...,p

and γp =(γ1, ..., γp)

′.

Denote βp = (β1, ..., βp)′ = Γ−1p γp, σ

2p = var(u2p+1 − β1u2p − ... − βpu21). Recall notation γ of

the smoothness parameter of ht in (2.2). Notation an << bn means an = o(bn). Notice that

Γ−1p exists5.

Theorem 2.1. (a) Let yt follow (2.1) where ht and ut satisfy Assumptions H and M. Suppose

that H satisfies

T 1/2 << H << T 1−(1/4γ). (2.11)

Then, for any p ≥ 1, the test statistics TS(u) and TS(u) corresponding to regression (2.9)

on squares u2t , have the following property

S(u) = S(u) + oP (1) = σ−2p β′pΓpβp + oP (1). (2.12)

(b) If {ut} is also an i.i.d. sequence, then for any p ≥ 1, βp = 0, and

TS(u) = TS(u) + oP (1)→D χ2p. (2.13)

This theorem shows, that the alternative H1 : βp 6= 0, is equivalent to γp 6= 0 and detected

with the rate T . Indeed, β′pΓpβp = γ ′pΓ−1p γp, and

β′pΓpβp ≥ ||βp||2λmin, γ ′pΓ−1p γp ≥ ||γp||2λ−1max,

where 0 < λmin ≤ λmax <∞ are the smallest and largest eigenvalues of Γp and ||βp|| denotes

the Euclidean norm of βp. It also shows that βp = Γ−1p γp in the ”true” value of βp estimated

by OLS in regression (2.9).

Recall that Assumption H is satisfied with γ = 1 by deterministic weights ht = g(t/T )

where g is a continuous piecewise differentiable function with a bounded derivative. Then

assumption (2.11) on bandwidth H becomes

T 1/2 << H << T 3/4. (2.14)

Test statistic TS(u) for ARCH effects in ut is based on regression (2.9) on squares u2t and

requires 6 finite moment of ut, see Assumption M. The same test based on the regression

|ut| = β0 + β1|ut−1|+ ...+ βp|ut−p|+ ηt (2.15)

allows to reduce the number of moments to E|ut|θ <∞, θ > 3. The following theorem shows

that the results of Theorem 2.1 remain valid with obvious corrections: TS(u) and TS(u)

5The existence of Γ−1p follows from Lemma 3.1(i) in Dalla, Giraitis, and Phillips (2020) because the sta-

tionary sequence {u2t} has spectral density. The latter follows from the absolute sumability of the covariance

function γk, see (6.52).

8

are computed using |ut| and |ut| instead of u2t and u2t , and in (2.12) σp, βp, Γp are defined

setting

γk = cov(|uk|, |u0|), σ2p = var(|up+1| − β1|up| − ...− βp|u1|). (2.16)

Theorem 2.2. The results of Theorem 2.2 remain valid for test statistics TS(u) and TS(u)

obtained from regression (2.15) on |ut| with σp, βp, Γp in (2.12) defined using (2.16).

In Assumption M, it suffices to assume E|ut|θ <∞, θ > 3 instead of θ > 6.

Similarly, testing for ARCH effects based on regression

ut = β0 + β1ut−1 + ...+ βput−p + ηt (2.17)

results in testing for the presence of correlation of ut at lags 1, ..., p. In such case, the

statistics TS(u) and TS(u) computed using ut and ut instead of u2t and u2t satisfy the results

of Theorem 2.1 with σp, βp, Γp in (2.12) defined correspondingly setting

γk = cov(uk, u0), σ2p = var(up+1 − β1up − ...− βpu1). (2.18)

Theorem 2.3. The results of Theorem 2.2 remain valid for test statistics TS(u) and TS(u)

obtained from regression (2.17) on ut with σp, βp, Γp in (2.12) defined using (2.18).

In Assumption M, it suffices to assume E|ut|θ <∞, θ > 3 instead of θ > 6.

Next we show that we can can test for absence of correlation in ut using the correlogram

of ut. This is an important step in data analysis where we do not know in advance whether

the series yt = htut we observe is a sequence of uncorrelated random variables. In particular,

before proceeding to testing for ARCH effects in ut, we may wish first to test for absence of

correlation in ut. Next theorem shows that such testing can be based on ut .

Denote for k = 0, 1, ...

ru,k = T−1∑T

t=k+1(ut − u)(ut−k − u), (2.19)

ru,k = T−1∑T

t=k+1(ut − Eut)(ut−k − Eut−k).

Theorem 2.4. Suppose that assumptions of Theorem 2.1 are satisfied. Then

ru,k = ru,k + oP (1) = cov(uk, u0) + oP (1), k ≥ 0. (2.20)

(b) In addition, if {ut} is an i.i.d. sequence then

T 1/2ru,k = T 1/2ru,k + oP (1)→ N (0, (Eu21)2) k ≥ 1. (2.21)

Moreover, in Assumption M it suffices to assume E|ut|θ <∞, θ > 3 instead of θ > 6.

9

Denote ρu,k = ru,k/ru,0, k = 0, 1, 2, .... If {ut} is an i.i.d. sequence then (2.21) of Theorem

2.4 implies that for any m = 1, 2, ...

T 1/2(ρu,1, ..., ρu,m)→D N (0, Im). (2.22)

This shows that using ut we can perform standard tests for absence of correlation at indi-

vidual lag k and cumulative Ljung-Box test at lag m as if i.i.d. variables ut were observed.

Similarly, it can be show that approximation |ut| ∼ |ut| and u2t ∼ u2t allows to use |ut|and u2t to test for absence of correlation in |ut| and u2t as if |ut| and u2t were observed. Testing

for correlation in |ut| requires E|ut|θ <∞, θ > 3 while testing for correlation in u2t requires

E|ut|θ <∞, θ > 6.

Notice that no ARCH effects in ut means no correlation in ut and u2t . This is a slightly

weaker assumption than the assumption of the i.i.d. property of ut. The later lead to

standard approximations (2.13) and (2.22), which are not guaranteed for non i.i.d. noises

ut.

3 Monte Carlo Analysis

In this section we present Monte Carlo findings on the finite sample performance of the test

for the presence of stationary volatility (ARCH effects) in artificially generated uncorrelated

data, given by

yt = htut, ut = σtεt, t = 1, ..., T. (3.1)

We evaluate the rejection frequency (in %) of the test statistics TS(u) for the presence of

a stationary non-constant volatility factor σt. We use sample sizes T = 200, 400, 800, 1600

and report testing results for the lag order, p = 5. Testing results for lags p = 1, 10 suggest

similar patterns to those discussed below, and are available upon request. Each experiment

involves 5000 replications. After estimating the persistent scale factor ht of volatility, the

test is applied on squares u2t = y2t /h2t , y

2t and for comparison on u2t . We expect similar

empirical size and power properties, when testing is based on either u2t or unobserved u2t ,

while the presence of ht in yt = htεt testing based on y2t leads to detection of spurious ARCH

effects. We report results for four kernel estimates h2t computed with bandwidth parameters

H = T 0.5, T 0.6, T 0.7, T 0.8.

Table 5.1 reports the size and power of the test for the ARCH model yt = ut = σtεt where

the persistent component ht is absent. This model allows to verify whether estimation of htintroduces any distortions to the empirical size and power of the standard test for ARCH

effects. For σt, ARCH(1) and GARCH(1,1) models are used:

σ2t = 1 + βu2t−1, β = 0, 0.2, 0.4, (3.2)

σ2t = 1 + 0.2u2t−1 + 0.7σ2

t−1. (3.3)

Except for the case of no ARCH effects, β = 0, such yt contains a stationary volatility

component σt which should result in detection of ARCH effects in the data.

10

From the table, it is clear that for β = 0 testing for ARCH effects based on the residuals u2tachieves the nominal size of α = 5% as both T and H increase and H meets the requirement

T 0.5 << H << T 0.75 of (2.11), and power is good throughout.

Table 5.2 reports the size and power of the test for yt as in (3.1) with a stochastic

persistent factor ht generated by a non-stationary ARFIMA(0, d, 0), d > 1 process I(d)t that

is re-scaled and bounded away from zero:

ht = T−(d−1/2)|I(d)t |+ 1, d = 1.2, 1.4, 1.5, (3.4)

and σt follows an ARCH(1) model (3.2) with β = 0, 0.2, 0.4. The case β = 0 corresponds to

yt = htεt satisfying the null hypothesis of no ARCH effect, i.e. σt = const.

We assume that {ht} and {ut} are mutually independent. Such ht satisfies Assumption

H with γ = d − 1/2 > 1/2, see Example 2.1. Here the importance and novelty of our new

testing procedure becomes clear. Specifically, we find that the standard ARCH test based

on y2t , which a priory assumes a constant ht, produces severe size distortions, under the null

hypothesis on no ARCH effects (β = 0). On the contrary, testing using the residuals u2tattains the nominal 5% size, as T increases and H satisfies the requirement T 0.5 << H <<

T 1−(1/4γ) for the range of H in (2.11). The test based on u2t tracks well the power of the test

based on u2t .

Table 5.3 reports testing results for yt as in (3.1) with a deterministic persistent scale

factor, given by

ht = sin(2πt/T ) + 2. (3.5)

σt follows either an ARCH(1) model (3.2) with β = 0, 0.2, 0.4 or a GARCH (1,1) model given

by (3.3). As before, the case β = 0 corresponds to yt = htεt satisfying the null hypothesis

of no ARCH effects which allows the evaluation of the size of the test. Such ht satisfies

Assumption H with γ = 1. This is the least favourable case for satisfactory performance of

the standard ARCH test. We find severe size distortions when such test is applied directly

on y2t . The test based on u2t approaches the nominal 5% size as long as H satisfies the range

condition T 0.5 << H << T 0.75 given in (2.11). Overall, the results of the test based on the

residuals u2t and y2t are similar to those in Table 5.3.

The above simulations are a small part of a much more extensive set of simulations. These

include considering tests for absence of ARCH effects based on untransformed residuals utand absolute values of the residuals |ut|, as well as considering a wide variety of deterministic

and stochastic settings for the persistent factor ht. All these results produce patterns that

are similar to those presented above and are available upon request. They are also presented

in detail in Chronopoulos (2021).

Some general conclusions follow from the simulation study. First we find that size and

power of the test for ARCH effects are not distorted by the estimation of ht, when htis constant. Second, the standard ARCH test is severely affected by the presence of a

persistent time varying scaling factor ht; it may suffer severe size distortions and produce

11

spurious power even for small sample size T . Overall, the new test on residuals u2t , produces

adequate nominal size and power throughout our experiments, illustrating the usefulness

and novelty of the approach.

4 Empirical Analysis

In this section we illustrate how the proposed testing methodology in Section 2 can be applied

to real data, to explore the presence or absence of stationary volatility component σt, besides

the persistent component, ht, in data yt presented as in (3.1). We examine the difference in

proportions (in %) of S&P 500 stock returns that test positive for ARCH effects when: i)

the random scale factor ht is not assumed constant, estimated by a kernel estimate ht, and

a scale robust series ut = h−1t yt is used in testing, and ii) ht is assumed constant (equal

to 1), i.e. the implicit assumption in regular ARCH testing. Further, we also explore one-

step-ahead volatility forecasting of var(yt|Ft−1) = h2tσ2t in order to assess how, forecasting

methods/models that that allow for the persistent non-stationary scaling factor ht, compare

to commonly used stationary alternatives for σ2t .

Our data is a panel of T = 1434 weekly observations on N = 505 S&P 500 stock

prices over the period 08/Jan/1993-27/Dec/20196. Following common practice we trim non–

working days, keep stocks that do not have missing values, and convert stock prices into

returns using log–differencing. After these operations, we retain T = 1340 observations of

N = 254 stocks. Further, we consider fitting a conditional mean model for stock returns,

and for that model we also require data on the S&P index6, data on the three Fama and

French (1993) factors and the risk free rate, available in French’s website.7

4.1 Testing for ARCH effect in empirical data

Following the literature, we assume that stock returns can be specified by the model

yit = µit + hituit, uit = σitε, µit = E[yit|Ft−1], var(yit|Ft−1) = h2itσ2it. (4.1)

Our aim is to discriminate between persistent, hit, and persistent and stationary volatility,

hitσit, by testing stock returns yit−µit for ARCH effects. Following testing setup in Section

2, for p = 1, 5, 10, we consider TS (ui), the test statistic based on u2it and TS (uit) based

on the squares u2it of the rescaled residuals, uit = h−1it (yit − µit), for every stock i, where

µit is an estimate of the conditional mean, and hit is the kernel estimate defined in (2.5),

for a range of pre-selected bandwidths H. We consider the full dataset, from 28/Oct/1994

to 27/Dec/2019, and the following three subsamples: i) 28/Oct/1994 – 28/Dec/2007, ii)

7/Jan/2000 - 30/Dec/2011 and iii) 7/Jan/2011–27/Dec/2019. We use these subsamples to

examine how the exclusion of highly volatility periods, like the 2008 financial crisis can affect

6Data are obtained from Bloomberg7http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data library.html

12

our testing results. Further, we examine how potential misspecification of the conditional

mean, µit = E(yit|Ft−1) 6= 0 and the inclusion of the estimate µit of conditional mean build

on covariates/factors that are known to proxy well for sensitivity to common risk factors

and capture strong common variation in stock returns, affect our testing for ARCH effects

results. The models we consider to estimate the conditional mean µt are the market factor

model and the three factor Fama and French (1993) model8. We report percentage of stocks

with ARCH effects across N = 254 stocks for the baseline model for conditional mean, µt = 0

and the two estimates of the conditional mean, see footnote 7.

Our empirical testing results are presented in Table 5.4. We report the proportion of

stocks with ARCH effects across the i stocks, for different bandwidths H, lags p and sub-

samples, for both the baseline model, µt = 0, where we do not estimate the conditional

mean µt and set it equal to 0, and where we use the market factor model and Fama-French

model to estimate it. Below, we provide a discussion on testing in subsamples and an overall

comment, for the baseline model, µt = 0, and the three factor Fama French model for µt.

The 2007 (28/Oct/1994 – 28/Dec/2007) subsample was generally a moderate period,

with the small outlier being the dotcom bubble. During this period we find that, in the

case where we do not estimate the conditional mean, i.e. µt = 0, the the proportion of

stock returns with ARCH effects, detected in the rescaled squared residuals u2t across stocks,

is considerably smaller than in the squared errors (yt − µt)2, i.e. a drop from 61.42% to

11.02%− 38.19 for different H’s.

The 2011–2019 (7/Jan/2011–27/Dec/2019) subsample, is the most moderate period we

consider. During this period, we still find a drop in proportion of stocks with ARCH effects

between the two tests, corresponding to the base line model, µt = 0, and Fama French factor

model for µt, from 33.86% to 1.57%− 18.50% across H. It is evident from comparisons with

other periods, that the proportion of stocks with ARCH effects when we do not estimate

the conditional mean is considerably smaller.

The 2000 -2011 (7/Jan/2000 - 30/Dec/2011) subsample, is more turbulent since it con-

tains the 2008 financial crisis. During this period, we find that for the baseline model, µt = 0,

the majority (83.47%) of the stocks test positive for ARCH effects in squares y2it. Using the

test on the rescaled residuals, we again, find that the proportion of stocks with ARCH effects

in squares u2it falls from 83.47% to 18.11%− 74.02 for different values of H. Similar drops in

the proportion of returns with ARCH effects in squares u2it is observed for the case where a

market factor or the Fama French factors are used to model the conditional mean.

The sample starting on 28/Oct/1994 and ending 27/Dec/2019, is the longest and most

turbulent period we have considered. We abstract from the impact of the CoViD-19 pan-

8The model with a conditional mean, can be written as yt = µt + htut with µt = rf,t + β1(Rm,t − rf,t)for the market factor model, and µt = rf,t + β1(Rm,t − rf,t) + β2SMBt + β3HMLt + htεt, for the three

factor Fama French model. Rm,t is the market factor, rf,t is the risk free rate and SMB, HML are the firm

size (small-minus-big) and book-to-market (high-minus-low) factors described in Fama and French (1993).

ht can be estimated using the errors yt = yt − µt of the two models, given estimates on βi, i = 1, 2, 3.

13

demic, since it is still not clear how it will affect the economy. During this period we find for

the baseline model, µt = 0, almost all (93.31%) of the stocks test positive for ARCH effects

in y2it, and similarly as above, the proportion stock returns with ARCH effects in the squares

u2t of rescaled residuals drops to 25.98%− 84.65% for different H values. Similar pattern is

observed for two our models for the conditional mean.

Across lags and subsamples, we find that the proportion of returns with ARCH effects

generally falls when testing the squares of the rescaled residuals and increases for the non-

rescaled ones. Further, accounting for conditional mean via the inclusion of covariates leads

to a further decrease of this proportion across sub-samples and lags, which is not an un-

expected result, since misspecification of the conditional mean, is known to affect ARCH

testing resulting is spurious ARCH effects. Overall, these empirical results are in line with

our Monte Carlo experiments. We treat these findings as evidence that stationary volatility

factor is overall considerably less pronounced in the data, than previously thought and that

further, the second moments of asset returns change slowly and persistently over time.

4.2 Volatility Forecasting

4.2.1 Forecasting Setup

Our in-sample empirical results suggest that stationary volatility is considerably less pro-

nounced in real data than what was previously thought. Specifically, we find that non-

stationary processes seem to co-drive the overall latent volatility. This can naturally have

implication for both the estimation and forecasting of volatility. In this section we explore

how this finding affects volatility forecasting, and we leave the case of estimation for future

work.

We assume that stock returns can be specified by the model

yt = htut, t = 1, . . . , T,

where ht is a persistent scale factor that satisfies Assumption H and ut is a stationary

sequence of uncorrelated random variables with appropriate moments. We re-write ut as

ut = σtεt (4.2)

to make explicit that σ2t follows some general stationary volatility process. We proceed to

produce volatility forecasts for a variety of model choices for σ2t . Specifically, we produce

1-step ahead forecasts of v2t = Et(y2t+1) via a recursive (expanding window) y1, ...., yt and

compare them with a volatility proxy r2t , discussed below.

In our forecasting exercise we consider three approaches to estimate and subsequently

forecast volatility. First, we set yt = σtεt, ht = 1, and fit the following commonly

used stationary models in the literature for σ2t . The models we consider are: ARCH(1),

GARCH(1, 1), the Glosten, Jagannathan, and Runkle (1993) GARCH, GJR(1, 1, 1), and

14

the random walk stochastic volatility (RW − SV ) of Ruiz (1994)9. Next, we set yt = htεt,

σ2t = 1, and use the estimator ht described in (2.5) to model volatility h2t non-parametrically.

Because it can be the case that both types of volatility can co-drive the overall latent volatil-

ity, we further use forecast combinations where at a first step we rescale the squared stock

returns y2s/h2s, 1 ≤ s ≤ t using the non-parametric estimate of volatility, hs, and subsequently

fit to them a parametric stationary model σ2s and produce forecasts σ2

t for σ2t . Finally we

use the latest value of the non-parametric estimate of volatility, h2t , and σ2t to produce the

overall volatility forecasts, v2t = h2t σ2t , v

2t = σ2

t and v2t = h2t in the third, first and the second

case.

Generally, we use the ARCH(1) model for performance check, since it is well known that

it does not give good forecasts especially for large T . Hence we also refrain from doing any

forecast combination with it, while we employ combinations of h2t with the rest of the models.

We use the subsamples y1, ..., yt, t ∈ [0.2T, T−1] to evaluate forecasting performance. Before

we present our results there are two important matters worthy of discussion.

First, because volatility is unobserved, the choice of a well-behaved proxy, to compare

with our forecast, is pertinent. Following Patton (2011), we use the squared demeaned

returns because they have been shown to be an unbiased proxy for volatility. We acknowledge

that while this is an unbiased proxy, it is noisy. This requires attention when the exercise

is about ranking different models in terms of their performance. Because the results are

subject to the proxy, this boils down in the choice of a ”robust” loss function, that can be

used in common tests, e.g. Diebold and Mariano (1995) or West (1996). Patton (2011) has

illustrated that the MSE loss function (among others), is reasonably robust. We use the

Diebold and Mariano (1995) test to rank the forecasting performance of the different models

considered.

Second, the estimation period of the initial volatility model parameters, i.e. the origin

0.2T where the recursive forecasting exercise starts can potentially be important in the

performance of the forecasting exercise, especially for the stationary models due to parameter

instability. There are arguments both for and against the importance of parameter instability

in forecasting models. Clements and Hendry (1996, 2000) among others, argue that ignoring

parameter instability is one of the main sources of forecast breakdowns, if not modelled

explicitly. On the other hand, Stock and Watson (1996) argue that there is very little

benefit in modelling this and Kim and Swanson (2014) argue that forecasts estimated by

recursive estimation performed as well or better than rolling windows forecasts, for a range

of models estimated from a large panel of macroeconomic time series.

9For the GJR-GARCH(1,1,1) which is an extension to GARCH(1, 1), σ2t = ω + β1σ

2t−1 + β2I(ut−1 <

0)u2t−1 +β3u2t−1. For the RW-SV model, σ2

t = exp (κt/2), where κt = ω+κt−1 + ηt, ηt ∼ N(0, 1) and ω ∈ R.

15

4.2.2 Results

In Table 5.5 we present the results from our main forecasting exercise where we use the

last decade of data as our sample, from 04/Jan/2008 to 27/Dec/2019. The table evaluates

forecasting performance of ARCH(1), GARCH(1), GJR(1, 1, 1), and RW − SW models,

kernel estimator h2t (k-method) and some forecast combinations. These are denoted as

k − GARCH(1, 1), k − GJR(1, 1, 1) and k − RWSV , where we omit the order for the

ARCH type models to save space in the table. We consider four values of H. In table

position (i, j) we report the proportion of stocks, where according to Diebold and Mariano

(1995) test, model i is significantly better at forecasting volatility than model j10.

Every j-th row relates to a particular volatility forecasting method j and elements of

the row report the proportion of stocks for which the method j significantly outperforms the

other methods according to the DM test. Similarly every j-th column reports the proportion

of stocks for which method j is significantly outperformed by the other methods according to

the DM test. Therefore, row and column averages are metrics for the forecasting performance

of the methods. Well performing methods have small column and large row averages and

vice versa.

The results from the forecasting exercise suggest that forecasts produced using the kernel

estimator outperform forecasts obtained from the other stationary model we considered.

Furthermore, using the kernel estimator to produce forecast combinations, in a two step

fashion, ameliorates the forecasting performance of stationary models. We can see that

across all models, k−GARCH produces the best forecasts, then comes kernel and k−GJRthat are followed by stationary alternatives.

5 Conclusion

Volatility modelling is at the core of financial econometrics where the majority of the litera-

ture, so far, focus on models that are stationary. Time variation in the conditional variance

comes from the use of either ARCH or SV types of processes. However, parameter esti-

mates near the boundary of stationarity suggest that ARCH/SV types of models cannot

easily accommodate the observed persistent variation in volatility estimates of macro/finance

datasets.

In this paper we contribute to the literature in three ways. Firstly, we provide a possible

setup for persistent processes, that can provide a superior approximation of the volatility

process. Secondly, we discuss how persistence allows the consistent estimation of unobserved

volatility, without strong parametric assumptions, and finally we provide a novel testing

10Specifically, if the Diebold and Mariano (2002) test statistic is below −1.96, we reject the null hypothesis

of equal forecasting performance of forecasting models at 5% significance level in favour of the alternative

that model i produces superior forecasts compared to model j. If test statistic above 1.96 we reject the null

in favour of the alternative that model j is significantly better at forecasting volatility than model i.

16

strategy that enable standard ARCH tests to separate persistent from stationary volatility

and discuss the conditions needed for this.

We provide Monte Carlo evidence, illustrating that the existence of a persistent scale

factor affects adversely standard ARCH tests, but not our newly proposed test.

We use our new testing scheme on U.S. stock returns and find, in line with our Monte

Carlo results, extensive support for the persistent volatility paradigm, suggesting that the

role of stationary time–varying volatility is not as outstanding in the data, as was previ-

ously thought and that further, the second moment of asset returns varies slowly over time.

Further, our forecasting exercise illustrates that this strategy has merit also in volatility

forecasting, either on its own or via forecasting combinations.

Concluding, our results, suggest a new avenue for testing for the presence of ARCH

effects in practice. Without using our new testing scheme, a typical ARCH test is applied

to some non-rescaled residuals, and then a researcher will proceed by fitting an ARCH(p)

or GARCH model, given a rejection of the null hypothesis. Using our proposed testing

strategy, the researcher needs to further run a second ARCH test on rescaled residuals. If

the new test cannot reject the null, then the rejection from the first test is spurious, since

it is the existence of a persistent scale factor ht that drives the volatility and not some

stationary process. In the case that the second test also rejects the null hypothesis of no

ARCH effects, then we need to inspect the kernel estimate for the presence of persistent

deterministic or random scale factor. If this appears constant across time, an ARCH(p)

or GARCH model should be fitted, whereas if it varies considerably in time, then a two-

step estimation is required. There are a number of avenues for future work, in particular,

developing a stability test for the persistent volatility component ht, similar to the work of

Chen and Hong (2012).

17

p = 5

T H data β = 0 β = 0.2 β = 0.4 Garch

200 T 0.5 ut 12.98 19.04 48.82 7.22

T 0.6 4.90 23.30 60.82 28.66

T 0.7 3.72 30.14 69.60 52.42

T 0.8 3.60 34.64 74.32 65.52

ut 4.34 40.30 78.94 77.12

yt 4.34 40.30 78.94 77.12

400 T 0.5 ut 13.18 42.80 89.26 33.86

T 0.6 4.80 52.00 94.12 76.14

T 0.7 3.62 60.04 96.08 90.56

T 0.8 3.80 64.70 97.14 94.94

ut 4.68 69.06 97.60 97.22

yt 4.68 69.06 97.60 97.22

800 T 0.5 ut 15.32 80.58 99.92 89.00

T 0.6 5.60 87.42 100.00 99.38

T 0.7 4.38 91.38 100.00 99.88

T 0.8 4.18 92.96 100.00 99.92

ut 4.74 94.14 100.00 99.98

yt 4.74 94.14 100.00 99.98

1600 T 0.5 ut 16.44 99.04 100.00 99.96

T 0.6 6.20 99.72 100.00 100.00

T 0.7 5.08 99.82 100.00 100.00

T 0.8 5.16 99.84 100.00 100.00

ut 5.44 99.86 100.00 100.00

yt 5.44 99.86 100.00 100.00

Table 5.1: Test for ARCH effects. Rejection frequencies (in %) at the 5% significance level

(β = 0 size, β > 0 power). Model: yt = σtεt where σ2t = 1 + βu2t−1; under GARCH

σ2t = 1 + 0.2u2t−1 + 0.7σ2

t−1; εt ∼ i.i.d.N (0, 1).

18

p = 5

T H data β = 0 β = 0.2 β = 0.4

d = 1.2 d = 1.4 d = 1.5 d = 1.2 d = 1.4 d = 1.5 d = 1.2 d = 1.4 d = 1.5

200 T 0.5 ut 11.20 11.80 12.14 19.12 19.48 19.38 49.26 49.00 49.14

T 0.6 3.90 4.06 4.30 23.40 23.30 23.36 62.06 61.40 61.52

T 0.7 3.92 3.82 3.68 31.40 30.00 29.70 71.22 70.86 70.70

T 0.8 6.32 5.04 4.78 38.68 37.14 36.54 77.26 76.54 76.38

ut 4.52 4.52 4.52 39.76 39.76 39.76 79.92 79.92 79.92

yt 22.58 20.24 19.36 56.08 54.18 53.76 84.12 83.46 83.34

400 T 0.5 ut 12.24 13.02 13.10 43.94 43.90 43.86 89.96 90.02 89.88

T 0.6 4.08 4.58 4.68 54.08 53.34 53.50 94.64 94.54 94.42

T 0.7 4.52 4.20 4.04 63.70 62.56 62.22 96.80 96.60 96.56

T 0.8 8.34 6.62 6.00 71.02 69.42 68.80 97.92 97.70 97.68

ut 5.12 5.12 5.12 70.10 70.10 70.10 97.84 97.84 97.84

yt 36.40 32.48 30.64 83.80 81.96 81.70 98.78 98.68 98.52

800

T 0.5 ut 14.22 15.06 15.30 81.62 81.34 81.34 99.92 99.92 99.92

T 0.6 4.86 5.22 5.38 88.68 88.30 88.22 99.98 99.98 99.98

T 0.7 4.66 4.32 4.52 92.58 92.10 91.94 99.98 99.98 99.98

T 0.8 9.46 6.28 5.74 95.22 94.10 93.82 99.98 99.98 99.98

ut 5.34 5.34 5.34 94.46 94.46 94.46 99.98 99.98 99.98

yt 51.20 45.04 43.28 98.02 97.68 97.46 99.98 99.98 99.98

1600

T 0.5 ut 15.44 15.94 16.10 99.28 99.26 99.24 100.00 100.00 100.00

T 0.6 5.32 5.64 5.92 99.72 99.68 99.66 100.00 100.00 100.00

T 0.7 5.76 5.14 5.06 99.86 99.84 99.80 100.00 100.00 100.00

T 0.8 11.98 7.16 6.22 99.92 99.90 99.88 100.00 100.00 100.00

ut 5.28 5.28 5.28 99.88 99.88 99.88 100.00 100.00 100.00

yt 65.84 57.46 55.22 99.98 99.96 99.96 100.00 100.00 100.00

Table 5.2: Test for ARCH effects. Rejection frequencies (in %) at the 5% significance level

(β = 0 size, β > 0 power). Model: yt = htut, ht = |I(d)t |/T d−1/2+1, I(d)t ∼ ARFIMA(0, d, 0),

ut = σtεt where σ2t = 1 + βu2t−1; εt ∼ i.i.d.N (0, 1).

19

p = 5

T H data β = 0 β = 0.2 β = 0.4 Garch

200 T 0.5 ut 10.50 17.96 49.14 8.02

T 0.6 3.20 24.42 62.84 34.96

T 0.7 7.64 40.68 74.92 66.44

T 0.8 25.10 60.12 84.08 83.34

ut 4.34 40.30 78.96 77.12

yt 73.66 86.78 93.62 95.22

400 T 0.5 ut 12.10 42.62 89.30 35.12

T 0.6 3.58 53.36 94.34 79.00

T 0.7 7.90 68.78 96.86 95.10

T 0.8 40.86 86.58 98.64 98.60

ut 4.68 69.04 97.60 97.22

yt 95.40 99.14 99.80 99.92

800 T 0.5 ut 14.42 80.62 99.92 89.58

T 0.6 4.74 88.16 100.00 99.52

T 0.7 7.78 93.88 100.00 99.96

T 0.8 60.80 98.98 100.00 100.00

ut 4.74 94.14 100.00 99.98

yt 99.98 100.00 100.00 100.00

1600 T 0.5 ut 15.96 99.08 100.00 99.96

T 0.6 5.68 99.72 100.00 100.00

T 0.7 7.06 99.84 100.00 100.00

T 0.8 80.06 100.00 100.00 100.00

ut 5.44 99.86 100.00 100.00

yt 100.00 100.00 100.00 100.00

Table 5.3: Test for ARCH effects. Rejection frequencies (in %) at the 5% significance level

(β = 0 size, β > 0 power). Model: yt = htut, ht = sin(2πt/T ) + 2, ut = σtεt where

σ2t = 1 + βu2t−1; under GARCH σ2

t = 1 + 0.2u2t−1 + 0.7σ2t−1; εt ∼ i.i.d.N (0, 1).

20

Testing for ARCH Effects Results

No conditional mean

H data p = 1 p = 5 p = 10

2007 2000-2011 2011-2019 2019 2007 2000-2011 2011-2019 2019 2007 2000-2011 2011-2019 2019

T 0.5 ut 11.02 18.11 1.57 25.98 6.30 6.69 1.57 13.39 5.91 5.12 3.54 11.02

T 0.55 16.54 28.35 3.54 42.52 8.27 14.96 2.36 36.61 7.09 11.42 2.36 30.32

T 0.6 18.90 37.80 7.48 61.02 14.17 29.92 4.72 64.57 12.21 24.41 5.51 59.45

T 0.7 29.92 66.54 16.54 81.50 31.89 72.05 18.90 87.80 31.50 70.47 18.90 90.16

T 0.75 38.19 74.02 18.50 84.65 46.85 81.89 27.17 91.73 46.85 83.07 29.92 94.49

yt 61.42 83.47 33.86 93.31 79.13 90.16 55.51 96.46 79.53 92.13 56.30 98.43

Market factor model for conditional mean

H data p = 1 p = 5 p = 10

2007 2000-2011 2011-2019 2019 2007 2000-2011 2011-2019 2019 2007 2000-2011 2011-2019 2019

T 0.5 ut 6.69 6.30 2.36 12.21 2.36 4.33 3.54 7.09 5.12 4.33 7.09 6.69

T 0.55 10.24 13.39 3.15 25.20 5.51 7.09 2.76 15.75 3.15 4.33 1.97 9.84

T 0.6 13.39 20.47 4.33 37.01 9.45 12.21 3.94 33.07 6.69 12.21 1.57 30.71

T 0.7 23.23 37.40 7.48 56.69 22.05 42.13 7.87 61.02 22.84 45.28 6.69 67.72

T 0.75 31.50 48.82 7.48 66.54 35.83 58.27 9.84 73.62 37.80 62.99 8.27 79.92

yt 60.63 74.02 21.65 84.25 77.95 83.07 19.69 93.70 79.13 88.98 22.84 94.09

Fama French model for conditional mean

H data p = 1 p = 5 p = 10

2007 2000-2011 2011-2019 2019 2007 2000-2011 2011-2019 2019 2007 2000-2011 2011-2019 2019

T 0.5 ut 9.45 16.14 0.79 23.23 5.12 5.51 3.15 11.02 5.91 4.33 2.36 9.84

T 0.55 14.57 26.77 3.15 38.58 7.87 13.39 1.97 33.86 5.12 9.45 1.18 29.92

T 0.6 18.50 36.61 6.69 57.87 10.63 29.53 3.94 62.21 9.06 22.84 4.72 58.27

T 0.7 27.17 62.60 13.39 79.53 29.92 68.11 15.75 85.83 28.74 68.50 16.54 88.58

T 0.75 35.04 72.05 16.54 82.68 43.70 78.35 24.02 90.16 44.09 79.53 24.02 93.70

yt 59.45 80.71 31.89 92.91 79.53 88.98 49.21 96.06 79.13 92.13 52.76 98.82

Table 5.4: Proportion of stock returns with ARCH effects in squares across the i stocks, for

different bandwidths H, lags p and subsamples defined in the main text. Testing at the 5%

significance level. Presented results are for baseline model, µt = 0, and for the cases where

the conditional mean µt is estimated using the Market factor and Fama-French models.

21

Method H ARCH GARCH k −GARCH kernel GJR k −GJR SV k − SV AV ERAGE (row)

ARCH T 0.50 - 0.04 0.00 0.01 0.05 0.02 0.01 0.05 0.02

T 0.55 0.00 0.01 0.01 0.02 0.02

T 0.60 0.00 0.02 0.02 0.02 0.02

T 0.65 0.00 0.03 0.02 0.02 0.02

T 0.70 0.00 0.03 0.04 0.02 0.02

GARCH T 0.50 0.54 - 0.02 0.04 0.09 0.02 0.10 0.14 0.12

T 0.55 0.02 0.05 0.05 0.12 0.12

T 0.60 0.02 0.07 0.07 0.11 0.13

T 0.65 0.03 0.14 0.09 0.10 0.14

T 0.70 0.04 0.22 0.15 0.10 0.15

k −GARCH T 0.50 0.72 0.46 - 0.14 0.33 0.11 0.32 0.43 0.31

T 0.55 0.71 0.47 0.19 0.33 0.15 0.34 0.34 0.32

T 0.60 0.69 0.43 0.27 0.31 0.19 0.30 0.29 0.31

T 0.65 0.69 0.36 0.36 0.25 0.23 0.23 0.23 0.29

T 0.70 0.63 0.31 0.50 0.23 0.28 0.18 0.18 0.29

Kernel T 0.50 0.67 0.37 0.00 - 0.30 0.04 0.30 0.37 0.26

T 0.55 0.63 0.39 0.00 0.28 0.07 0.29 0.27 0.24

T 0.60 0.57 0.31 0.00 0.25 0.08 0.23 0.18 0.20

T 0.65 0.48 0.22 0.00 0.18 0.07 0.13 0.10 0.15

T 0.70 0.37 0.19 0.01 0.16 0.07 0.07 0.07 0.12

GJR T 0.50 0.53 0.17 0.03 0.04 - 0.05 0.13 0.17 0.14

T 0.55 0.03 0.05 0.08 0.14 0.14

T 0.60 0.04 0.09 0.08 0.13 0.15

T 0.65 0.04 0.15 0.13 0.14 0.16

T 0.70 0.04 0.25 0.17 0.14 0.18

k −GJR T 0.50 0.59 0.32 0.00 0.04 0.24 - 0.21 0.26 0.21

T 0.55 0.57 0.30 0.00 0.06 0.25 0.20 0.20 0.20

T 0.60 0.53 0.25 0.00 0.09 0.18 0.17 0.14 0.17

T 0.65 0.43 0.20 0.00 0.13 0.15 0.09 0.09 0.14

T 0.70 0.35 0.17 0.02 0.15 0.12 0.07 0.06 0.12

SV T 0.50 0.46 0.21 0.03 0.05 0.17 0.03 - 0.23 0.15

T 0.55 0.03 0.05 0.04 0.18 0.14

T 0.60 0.04 0.07 0.06 0.17 0.15

T 0.65 0.05 0.13 0.09 0.17 0.16

T 0.70 0.07 0.24 0.13 0.13 0.18

k − SV T 0.50 0.46 0.22 0.02 0.04 0.18 0.02 0.10 - 0.13

T 0.55 0.50 0.22 0.02 0.05 0.17 0.05 0.15 0.15

T 0.60 0.50 0.21 0.04 0.09 0.17 0.07 0.14 0.15

T 0.65 0.47 0.22 0.05 0.17 0.16 0.09 0.13 0.16

T 0.70 0.46 0.21 0.06 0.28 0.15 0.14 0.11 0.18

AV ERAGE T 0.50 0.50 0.22 0.01 0.05 0.17 0.04 0.15 0.21

(column) T 0.55 0.49 0.22 0.01 0.06 0.17 0.06 0.15 0.16

T 0.60 0.48 0.20 0.02 0.09 0.15 0.07 0.14 0.13

T 0.65 0.45 0.18 0.02 0.14 0.13 0.09 0.10 0.11

T 0.70 0.42 0.16 0.03 0.21 0.12 0.12 0.08 0.09

Table 5.5: In table position (i, j) we report the proportion of stock returns, where Diebold and

Mariano (1995) test rejects at the 5% significance level the equal performance of forecasting models

i and j in favour of the alternative that the model i is significantly better at forecasting volatility

than model j. Row and column averages are metrics for the forecasting performance of the models.

The best performing model is the one with the smallest column and highest row average.

22

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25

6 Appendix

6.1 Proof of Theorems 2.1, 2.2 and 2.4.

In the proof of Theorem 2.1 we use the claim that h2t satisfies Assumption H which is verified

the next proposition.

Proposition 6.1. If ht, t = 1, ..., T satisfies Assumption H with parameters γ and α then

h2t , t = 1, ..., T satisfies Assumption H with parameters γ and α/2.

Proof of Proposition 6.1 By assumption, ht satisfies (2.2),

|ht − hj| ≤ (|t− j|/T )γξtj, t, j = 1, ...., T

and ξtj, ht satisfies (2.3). Therefore

|h2t − h2j | ≤ |ht − hj| |ht + hj| ≤ (|t− j|/T )γξ∗tj, ξ∗tj = ξtj(|ht|+ |hj|).

Condition (2.3) implies that ξ∗tj and h2t satisfy (2.3) with parameter α/2. 2

Proof of Theorem 2.1. In Theorem 2.1 we analyse the Wald version of test for H0

hypothesis for absence of ARCH effects ut. First we recall definitions of S(u) and S(u).

Given data u = [u1, u2, . . . , uT ], we define the test statistic for testing H0 as follows:

S(u) = σ−2β′p(X

′X)βp, βp = (X ′X)−1X ′Y, σ2p = (Y −Xβp)′(Y −Xβp) (6.1)

where u2 = T−1∑T

t=1 u2t , Y is (T − p)× 1 vector and X is (T − p)× 1 design matrix:

Y = (u2p+1 − u2, ..., u2T − u2)′,

X =

x1,1 x1,2 ... x1,px2,1 x2,2 ... x2,p... ... ... ...

xT−p,1 xT−p,2 ... xT−p,p

=

u2p − u2 u2p−1 − u2 ... u21 − u2u2p+1 − u2 u2p − u2 ... u22 − u2

... ... ... ...

u2T−1 − u2 u2T−2 − u2 ... u2T−p − u2

.

Here βp denotes the estimated regression coefficients of u2t−1, . . . , u2t−p in a regression of u2t on

a constant and u2t−1, . . . , u2t−p. Similarly, let βp denote the estimated regression coefficients

of u2t−1, . . . , u2t−p in a regression of u2t on a constant and u2t−1, . . . , u

2t−p. Then

S(u) = σ−2p β′p(X

′X)βp, βp = (X ′X)−1X ′Y , σ2p = (Y − Xβp)′(Y − Xβp) (6.2)

where u2 = T−1∑T

t=1 u2t ,

Y = (u2p+1 − u2, ..., u2T − u2)′,

X =

x1,1 x1,2 ... x1,px2,1 x2,2 ... x2,p... ... ... ...

xT−p,1 xT−p,2 ... xT−p,p

=

u2p − u2 u2p−1 − u2 ... u21 − u2u2p+1 − u2 u2p − u2 ... u22 − u2

... ... ... ...

u2T−1 − u2 u2T−2 − u2 ... u2T−p − u2

.

26

Observe that we can write

T−1(X ′X) = (gij)i,j=1,...,p, T−1(X ′Y ) = (g0j)j=1,...,p, where (6.3)

gij = T−1∑T

t=p+1(u2t−i − u2)(u2t−j − u2).

Similarly,

T−1(X ′X) = (gij)i,j=1,...,p, T−1(X ′Y ) = (g0j)j=1,...,p, where (6.4)

gij = T−1∑T

t=p+1(u2t−i − u2)(u2t−j − u2).

Proof of Theorem 2.1 is based on the Lemmas 6.1 and 6.2 below. Auxiliary results used to

prove these lemmas are placed in Section 6.2. Denote

γk = T−1T∑

t=k+1

(u2t − Eu2t

)(u2t−k − Eu2t−k

)(6.5)

the autocovariance function of squared residuals u2t and of u2t , respectively, where u2 =

T−1∑T

t=1 u2t . Denote Γp = (γ|i−j|)i,j=1,...,p, γp = (γ1, ..., γp)

′, βp = Γ−1p γp.

To prove the theorem we will derive the following results:

T−1(X ′X) = Γp + oP (1), T−1(X ′Y ) = γp + oP (1), (6.6)

T−1(X ′X) = Γp + oP (1), T−1(X ′Y ) = γp + oP (1), (6.7)

Γp →p Γp, γp →P γp, βp → βp, (6.8)

T−1σ2p →P σ

2p, T−1σ2

p →P σ2p, (6.9)

where βp = Γ−1p γp, Γp =(γ|j−k|

)j,k=1,...,p

, γp =(γ1, ..., γp)

′, γk = cov(u2k, u20), k ≥ 0, and

σ2p = var(u2p+1 − β1u2p − ...− βpu21).

In addition, if {ut} is an i.i.d. sequence, then

T−1/2(X ′Y ) = T−1/2γp + oP (1), (6.10)

T−1/2(X ′Y ) = T−1/2γp + oP (1), (6.11)

T−1/2γp → N (0, Ipγ0), γ0 = var(u21). (6.12)

In view of (6.3) and (6.4), relations (6.6), (6.7) and (6.8) follow from (6.13), (6.14) and (6.15)

of Lemma 6.1, respectively, while (6.10) and (6.11) follow from (6.16) and (6.17) of Lemma

6.1. Convergence (6.12) for i.i.d. r.v. u2t − Eu2t is a well known.

Convergence (6.9) follows from the definitions of σ2p and σ2

p, (6.6)-(6.9), noting that

T−1Y ′Y = g00 →P γ0, T−1Y ′Y = g00 →P γ0 and using the equality σ2

p = γ0 − 2γ ′pβp +

β′pΓpβp.

Applying in S(u) and S(u), given by (6.2) and (6.1), relations (6.6)-(6.9), we obtain

βp →P βp = Γ−1p γp, βp →P βp,

27

S(u) = S(u) + oP (1) = σ−2p β′pΓpβp + oP (1)

which proves the claim (2.12) of the theorem.

In addition, if {ut} is an i.i.d. sequence, then it holds Γp = γ0Ip and σ2p = γ0. Then,

using (6.10)–(6.12), we obtain

σ−2p (X ′X) = γ−20 Ip(1 + oP (1)), σ−2p (X ′X) = γ−20 Ip(1 + oP (1)),

T−1/2βp →D γ1/20 N (0, Ip), T−1/2βp →D γ

1/20 N (0, Ip).

This together with definitions of S(u) and S(u) implies

TS(u) = TS(u) + oP (1)→D χ2p,

which proves (2.13). This completes the proof of the theorem. 2

Proof of Theorems 2.2-2.3. It follows using the same arguments as in the proof of

Theorem 2.1. 2

Proof of Theorem 2.4. The claims (2.20) and (2.21) follow using the same argument as in

the proof of (6.21) and (6.22) of Lemma 6.2 noting that convergence T 1/2ru,k → N (0, (Eu21)2)

is well known for i.i.d. r.v.’s. 2

Lemma 6.1. (a) Suppose that (ht, ut) satisfy Assumptions M and H, and the bandwidth H

satisfies (2.11). Then for i, j = 0, 1, ..., p,

gij = γ|i−j| + oP (1), (6.13)

gij = γ|i−j| + oP (1), (6.14)

γk →p γk, k ≥ 0. (6.15)

(b) In addition, if {ut} is an i.i.d. sequence, then for j = 1, ..., p,

T 1/2g0j = T 1/2γj + oP (1), (6.16)

T 1/2g0j = T 1/2γj + oP (1). (6.17)

Proof of Lemma 6.1.

Proof of (6.13). It suffices to verify (6.13) for i ≤ j. Denote

γk = T−1T∑

t=k+1

(u2t − u2

)(u2t−k − u2

), k = 0, 1, 2, .... (6.18)

Then setting k = j − i, we can write

gij = T−1T−i∑

t=p+1−i

(u2t − u2)(u2t−k − u2) = T−1T−i∑

t=k+1+(p−j)

(u2t − u2)(u2t−k − u2)

28

= γ|j−i| − δij, δij = T−1[

k+p−j∑t=k+1

+T∑

t=T−i+1

](u2t − u2)(u2t−k| − u2).

So,

gij = γ|i−j| + (γ|i−j| − γ|i−j|)− δij. (6.19)

By (6.21) of Lemma 6.2, γ|i−j| − γ|i−j| = oP (1). On the other hand, straightforward use of

(6.45) and (6.47) of Lemma 6.4 implies

δij = OP (T−1) (6.20)

which together with (6.19) proves (6.13): gij = γ|i−j| + oP (1).

Proof of (6.14). Property (6.14) follows from the proof of (6.13) setting ht = ht which implies

u2t = u2t .

Proof of (6.15). By assumption, ut is a stationary ergodic sequence and Eu6t < ∞. Then

sequence zt =(u2t − Eu2t

)(u2t−k − Eu2t−k

)is stationary and ergodic with E|zt| < ∞ which

implies γk →P Ezk = cov(u2k, u20) = γk.

Proof of (6.16). By (6.19), (6.20) and (6.22), for j = 1, ..., p,

T 1/2g0j = T 1/2γj + oP (1).

Proof of (6.17). Property (6.17) follows from (6.16) setting ht = ht. 2

Lemma 6.2. (a) Suppose that (ht, ut) satisfy Assumptions M and H, and the bandwidth H

satisfies (2.11). Then

γk − γk = oP (1), k ≥ 0. (6.21)

(b) In addition, if {ut} is an i.i.d. sequence then

γk − γk = oP (T−1/2), k ≥ 1. (6.22)

Proof of Lemma 6.2. Denote

γ∗k = T−1T∑

t=k+1

(u2t − Eu2t

)(u2t−k − Eu2t−k

), k = 0, 1, 2, .... (6.23)

Then

γk − γk = (γ∗k − γk) + (γk − γ∗k).

Thus, to prove (6.21), it suffices to show

γ∗k − γk = oP (1), k ≥ 1, (6.24)

γk − γ∗k = oP (1), k ≥ 0, (6.25)

29

γ∗0 − γ0 = oP (1). (6.26)

In turn, to prove (6.22), we show in addition that for an i.i.d. sequence {ut} it holds

γ∗k − γk = oP (T−1/2), k ≥ 1, (6.27)

γk − γ∗k = oP (T−1/2), k ≥ 1. (6.28)

Proof of (6.24). Recall, that by assumption, Eu2t = 1. We have,

(u2t − Eu2t )(u2t−k − Eu2t−1)− (u2t − Eu2t )(u2t−k − Eu2t−k)= {(u2t − u2t ) + (u2t − 1)}{(u2t−k − u2t−k) + (u2t−k − 1)} − (u2t − 1)(u2t−k − 1)

= (u2t − u2t )(u2t−k − u2t−k) + (u2t − u2t )(u2t−k − 1) + (u2t − 1)(u2t−k − u2t−k).

Hence,

γ∗k − γk = ST,1 + ST,2 + ST,3, (6.29)

where

ST,1 = T−1∑T

t=k+1(u2t − u2t )(u2t−k − u2t−k), ST,2 = T−1

∑Tt=k+1(u

2t − u2t )(u2t−k − 1),

ST,3 = T−1∑T

t=k+1(u2t − 1)(u2t−k − u2t−k). (6.30)

To prove (6.24), it remains to show that ST,` in (6.30) satisfy

ST,` = oP (1), ` = 1, 2, 3. (6.31)

Notice that

u2t − u2t = (h2t h−2t − 1)u2t . (6.32)

We have

|ST,1| ≤ T−1|T∑

t=k+1

|(h2t h−2t − 1)(h2t−kh−2t−k − 1)|u2tu2t−k

≤ ( maxt=1,...,T

|h2t/h2t − 1|)2 sT , sT = T−1T∑

t=k+1

u2tu2t−k.

By (6.39) of Lemma 6.3, maxt=1,...,T |h2t/h2t − 1| = oP (1). By Assumption M, {ut} is a

stationary sequence, and EsT = E[u21u21−k]] ≤ E[u41] < ∞. Therefore, sT = OP (1). This

implies

ST,1 = oP (1).

The proof of (6.31) for ST,2, ST,3 is similar to that for ST,1. This completes the proof of

(6.24).

30

Proof of (6.25). Recall, that by assumption, Eu2t = 1. We have,

(u2t − u2)(u2t−k − u2)− (u2t − 1)(u2t−k − 1)

= {(u2t − 1) + (1− u2)}{(u2t−k − 1) + (1− u2)} − (u2t − 1)(u2t−k − 1)

= (u2 − 1)2 + (1− u2)(u2t−k − 1) + (u2t − 1)(1− u2).

Notice that

T−1∑T

t=k+1(u2t − 1) = (u2 − 1)− T−1

∑kt=1(u

2t − 1),

T−1∑T

t=k+1(u2t−k − 1) = (u2 − 1)− T−1

∑Tt=T−k+1(u

2t − 1).

Hence,

γk − γ∗k = T−1∑T

t=k+1{(u2 − 1)2 − (u2 − 1)(u2t−k − 1)− (u2t − 1)(u2 − 1)}= (u2 − 1)2{T−1(T − k)− 2)

+(u2 − 1){T−1∑k

t=1(u2t − 1) + T−1

∑Tt=T−k+1(u

2t − 1)}. (6.33)

By Assumption M, {ut} is a stationary α-mixing sequence. Applying in (6.33) the bounds

(6.45) and (6.46) of Lemma 6.4, we obtain (6.25):

γk − γ∗k = oP (1), k ≥ 0.

Proof of (6.26) By Assumption M, {ut} is a stationary α-mixing sequence. Using equality

a2 − b2 = (a− b)2 + (a− b)2b with a = u2t − 1, b = u2t − 1 we obtain

γ∗0 − γ0 = T−1∑T

t=1{(u2t − 1)2 − (u2t − 1)2}= T−1

∑Tt=1{(u2t − u2t )2 + (u2t − u2t )2(u2t − 1)}.

Recall that u2t−u2t = (h2t/h2t−1)u2t , and by (6.39) of Lemma 6.3, iT := maxt=1,...,T |h2t/h2t−1| =

oP (1). Therefore,

|γ∗0 − γ0| ≤ i2T(T−1

T∑t=1

u4t)

+ 2iT(T−1

T∑t=1

u2t |u2t − 1|)

= oP (1)qT , qT = T−1T∑t=1

(u4t + u2t ).

Since EqT = Eu41 +Eu21 <∞ this implies qT = OP (1) which proves (6.26): γ∗0 − γ0 = oP (1).

Proof of (6.27). By (6.29), γ∗k − γk = ST,1 + ST,2 + ST,3. To prove (6.27), it remains to show

that

ST,` = oP (T−1/2), ` = 1, 2, 3. (6.34)

First we evaluate ST,1. Using (6.32), we can bound

|ST,1| ≤ T−1|T∑

t=k+1

|(h2t h−2t − 1)(h2t−kh−2t−k − 1)|u2tu2t−k

31

≤ ( maxt=1,...,T

h−2t )2S ′T,1, S ′T,1 = T−1T∑

t=k+1

∣∣(h2t − h2t )(h2t−k − h2t−k)∣∣u2tu2t−k.By Lemma 6.3, maxt=1,...,T h

−2t = Op(1). Moreover, by (6.58) of Lemma 6.5,

E[∣∣(h2t − h2t )(h2t−k − h2t−k)∣∣u2tu2t−k]≤ (E[(h2t − h2t )2u2tu2t−k])1/2(E[(h2t−k − h2t−k)2u2tu2t−k])1/2

≤ C((H/T )2γ +H−1

)where C does not depend on t,H, T . Hence,

ES ′T,1 ≤ C((H/T )2γ +H−1

)= o(T−1/2)

under assumption (2.11) on H. So, S ′T,1 = oP (T−1/2). This proves (6.34) for ST,1.

Next we evaluate ST,2. Write,

ST,2 = T−1T∑

t=k+1

(h2t h−2t − 1)ζt, ζt = u2t (u

2t−k − 1). (6.35)

Write

h2t = h2t + (h2t − h2t ) = h2t (1 + xt), xt =h2t − h2th2t

.

Then

h2t

h2t=

1

1 + xt= 1− xt +

x2t1 + xt

= 1− xt +h2t

h2tx2t = 1− xt +

(h2t − h2t )2

h2th2t

.

Then, by (6.35),

ST,2 = −T−1T∑

t=k+1

xtζt + T−1T∑

t=k+1

(h2t − h2t )2

h2th2t

ζt

= : qT,1 + qT,2.

To prove (6.34) for ST,2, we verify that

qT,` = oP (T−1/2), ` = 1, 2. (6.36)

In (6.56) of Lemma 6.5 it is shown E|qT,1| = O((H/T )2γ+H−1

)= o(T−1/2) under assumption

(2.11) which proves (6.39) for qT,1.

32

Next, we bound

|qT,2| ≤ ( maxt=1,...,T

h−2t )( maxt=1,...,T

h−2t ) vT vT = T−1T∑

t=k+1

(h2t − h2t )2|ζt|.

By Assumption H, ht ≥ a > 0 a.s., and by (6.38) of Lemma 6.3, maxt=1,...,T h−2t = OP (1),

whereas using (6.57) of Lemma 6.5, we obtain

EvT ≤ T−1T∑

t=k+1

E[(h2t − h2t )2|ζt|] = O((H/T )2γ +H−1

).

This implies

qT,2 = OP

((H/T )2γ +H−1

)= oP (T−1/2)

under assumption (2.11) which proves (6.39). This verifies (6.34) for ST,2. The proof of

(6.34) for ST,3 is similar to the proof for ST,2. This completes the proof of (6.27).

Proof of (6.28) Using in (6.33) the bounds (6.48) and (6.46) of Lemma 6.4 we obtain

γk − γ∗k = Op

((H/T )γ +H−1/2)2

)+Op

(((H/T )γ +H−1/2)T−1

)= oP (T−1/2)

under assumption (2.11) on H. This proves (6.28) and completes the proof of the Lemma

6.2. 2.

Lemma 6.3. Suppose that (ht, ut) satisfy Assumptions M and H, and the bandwidth H sat-

isfies (2.11). Then there exists δ > 0 such that

maxt=1,...,T

|h2t − h2t | = oP (T−δ), (6.37)

maxt=1,...,T

h−2t = OP (1), (6.38)

maxt=1,...,T

|h2t/h2t − 1| = oP (T−δ). (6.39)

Proof of Lemma 6.3. Proof of (6.37). We have

h2t − h2t = K−1t∑T

j=1 bH,|t−j|(h2t − h2ju2j) (6.40)

= K−1t∑T

k=1 bH,|t−j|(h2t − h2j)u2j −K−1t h2t

∑Tj=1 bH,|t−j|(u

2j − 1).

Properties (2.6) of the kernel function K imply

maxt=1,...,T

K−1t ≤ CH−1 (6.41)

where C does not depend on T . By Assumption H and Proposition 6.1,

|h2t − h2j | ≤ (|t− j|/T )γξtj = (H/T )γ(|t− j|/H)γξtj

33

where {ξtj} and {h2t} satisfy the condition (2.3) of finite exponential moment with parameter

α/2 > 0. From (6.40) it follows

|h2t − h2t | ≤ (H/T )γ(maxj,t=1,...,T |ξtj|){maxt=1,...,T K−1t

∑Tj=1 bH,|t−j|(|t− j|/H)γu2j}

+(maxt=1,...,T h2t ){maxt=1,...,T

∣∣∑Tj=1 bH,|t−j|(u

2j − 1)

∣∣}. (6.42)

By (v) of Lemma C1 in the online supplement of Dendramis, Giraitis and Kapetanios (2021),

under Assumption H h2t and ξtk satisfy

max1≤t≤T

h2t = OP

((log T )2/α)

), max

1≤t,j≤T|ξtj| = OP

((log T )2/α)

). (6.43)

By Assumption M, {uj} is a stationary α-mixing sequence, and E|u1|θ <∞ for some θ > 6.

Then ξj = u2j−1 is also a stationary α-mixing sequence which satisfies α-mixing Assumption

M , see Theorem 14.1 in Davidson (1994), and E|ξ1|θ′<∞ where θ′ = θ/2 > 3. Under these

assumptions, Corollary 6(b) of Dendramis, Giraitis and Kapetanios (2018) implies that for

any ε > 0,

max1≤t≤T

∣∣H−1 T∑j=1

bH,|t−j|(ξj − Eξj)∣∣ = OP

(H−1/2

√log T + (HT )1/θ

′Hε−1). (6.44)

For H ≥ T 1/2 and θ′ > 3, the r.h.s. of (6.44) is of order oP (T−δ) for some δ > 0 when ε is

selected sufficiently small.

Setting b∗H,|t−j| = bH,|t−j|(|t− j|/H)γ, we have

K−1t∑T

j=1 b∗H,|t−j|u

2j = K−1t

∑Tj=1 b

∗H,|t−j|(u

2j − 1) +K−1t

∑Tj=1 b

∗H,|t−j|,

where under assumption (2.6), maxt=1,...,T K−1t

∑Tj=1 b

∗H,|t−j| = O(1). Then, using similar

argument as in the proof of (6.44), it follows that

maxt=1,...,T K−1t

∣∣∑Tj=1 b

∗H,|t−j|(u

2j − 1)

∣∣ = oP (T−δ).

Applying these bound in (6.42) we obtain

maxt=1,...,T

|h2t − h2t | = OP

((log T )1/α

){OP

((H/T )γ

)+ oP (T−δ)} = oP (T−δ

′)

for some δ′ > 0 under assumption (2.11) on H. This proves (6.37).

Finally, by Assumption H, ht ≥ a > 0 a.s. Thus,

mint=1,...,T

h2t = mint=1,...,T

(h2t − (h2t − h2t )

)≥ min

t=1,...,Th2t − max

t=1,...,T|h2t/h2t − 1| ≥ a− oP (T−δ)

which proves (6.38. In turn, (6.38) and (6.37) imply (6.39):

maxt=1,...,T

|h2t/h2t − 1| ≤ ( maxt=1,...,T

h−2t )( maxt=1,...,T

|h2t − h2t |) = oP (T−δ).

This completes the proof of the lemma. 2

34

6.2 Auxiliary lemmas

This section contains auxiliary lemmas used on the proofs of Section 6.1.

Lemma 6.4. (a) Under assumptions of Lemma 6.2(a), for any fixed k ≥ 1,

u2 − 1 = oP(1), (6.45)∑k

t=1 |u2t − 1| = Op(1),∑T

t=T−k+1 |u2t − 1| = Op(1). (6.46)∑kt=1(u

2t + u4t ) = Op(1),

∑Tt=T−k+1(u

2t + u4t ) = Op(1). (6.47)

(b) In addition, if ut is an i.i.d. sequence, then

u2 − 1 = OP

((H/T )γ +H−1/2

). (6.48)

Proof of Lemma 6.4. Proof of (6.45). We have

u2 − 1 = T−1T∑t=1

(u2t − 1) = T−1T∑t=1

(u2t − u2t ) + T−1T∑t=1

(u2t − 1)

=: Q1,T +Q2,T . (6.49)

We will show

Q1,T = oP (1), Q2,T = OP (T−1/2) (6.50)

which proves (6.45). We have,

|Q1,T | = T−1∣∣∣ T∑t=1

(h2t/h2t − 1)u2t

∣∣∣ ≤ ( maxt=1,...,T

|h2t/h2t − 1|)T−1T∑t=1

u2t (6.51)

= oP (1)T−1T∑t=1

u2t = oP (1),

by (6.39) of Lemma 6.3, noting that E(T−1∑T

t=1 u2t ) = Eu21 implies T−1

∑Tt=1 u

2t = OP (1).

Under Assumption M, by Conclusion 2.2 in Davydov (1968) (for more details see (A.11)

in Dendramis et al. (2021)), stationary α-mixing sequence zt = u2t − 1 has property

∞∑k=0

|cov(zk, z0)| <∞. (6.52)

Therefore,

EQ22,T = E(T−1

T∑t=1

(u2t − 1))2 = T−2T∑

k,j=1

cov(zk, zj)

35

≤ T−1∞∑

k=−∞

|cov(zk, z0)| ≤ CT−1

which proves the second claim in (6.50). This completes the proof of (6.45).

Proof of (6.46). We have

k∑t=1

|u2t − 1| =k∑t=1

|h2t/h2t − 1|u2t ≤ ( maxt=1,...,T

|h2t/h2t − 1|)k∑t=1

u2t = oP (1)

by (6.39) of Lemma 6.3 noting that for any fixed k,∑k

t=1 u2t = OP (1). This proves the first

claim in (6.46). The proof of the second claim is similar.

Proof of (6.47). We can bound

u2t + u4t = {(u2t − 1) + 1}+ {(u2t − 1) + 1}2 ≤ |u2t − 1|+ 1 + 2((u2t − 1)2 + 1).

Then (6.47) follows by the same argument as in the proof of (6.46).

Proof of (6.48). In view of (6.49) and (6.50), it suffices to show

Q1,T = Op

((H/T )γ +H−1/2

). (6.53)

By (6.51),

|Q1,T | ≤ ( maxt=1,...,T

h−2t )dT , dT = T−1T∑t=1

|h2t − h2t |u2t . (6.54)

By Lemma 6.3, maxt=1,...,T h−2t = OP (1). On the other hand, by (6.55) of Lemma 6.5,

E(h2t − h2t )2 ≤ C((H/T )2γ +H−1) where C does not depend on t,H, T , which implies

EdT ≤ T−1T∑t=1

(E(h2t − h2t )2

)1/2(Eu4t )

1/2

≤ C((H/T )2γ +H−1)1/2T−1T∑t=1

(Eu41)1/2 = C((H/T )2γ +H−1)1/2.

Hence dT = OP

((H/T )γ +H−1/2)

)which together with (6.54) proves (6.53). This completes

the proof of the lemma. 2

Lemma 6.5. (a) Under assumptions of Lemma 6.2(a),

E(h2t − h2t )2 ≤ C((H/T )2γ +H−1

). (6.55)

where C does not depend on t,H, T .

36

(b) If in addition, ut is a sequence of i.i.d. random variables, then for any k ≥ 1,

E∣∣∣T−1∑T

t=k+1 h−2t (h2t − h2t )ζt

∣∣∣ = O((H/T )2γ +H−1

), (6.56)

E[(h2t − h2t )2|ζt|] ≤ C((H/T )2γ +H−1

), (6.57)

E[(h2t−s − h2t−s)2u2tu2t−k] ≤ C((H/T )2γ +H−1

), for s = 0, k (6.58)

where ζt = u2t (u2t−k − 1) and C does not depend on t,H, T .

Proof of Lemma 6.5.

Proof of (6.55). Denote h2t = K−1t∑T

j=1 b|t−j|h2j . Then

h2t − h2t = (h2t − h2t ) + (h2t − h2t ), (6.59)

(h2t − h2t )2 ≤ 2(h2t − h2t )2 + 2(h2t − h2t )2.

We will show that

E(h2t − h2t )2 ≤ C(H/T )2γ, (6.60)

E(h2t − h2t )2 ≤ CH−1, (6.61)

which together with (6.59) proves (6.55).

Proof of (6.60). We have

E(h2t − h2t )2 = E(K−1t

∑Tj=1 b|t−j|(h

2t − h2j)

)2≤ K−2t

∑Tj,k=1 b|t−j|b|t−k|E[(h2t − h2j)(h2t − h2k)]

≤ CK−2t∑T

j,k=1 b|t−j|b|t−k|(E(h2t − h2j)2E(h2t − h2k)2

)1/2.

By Assumption H and Proposition 6.1, E(h2t − h2j)2 ≤ C((t − j)/T )γ where C does not

depend on t, j, T . Properties (6.41) and (2.6) of Kt and bt imply

maxt=1,...,T

K−1t

T∑j=1

b|t−j|(|t− j|H

)γ ≤ C. (6.62)

So,

E(h2t − h2t )2 ≤ C(H/T )2γ(K−1t

T∑j=1

b|t−j|(|t− j|/H)γ)2 ≤ C(H/T )2γ

where C does not depend on t,H. This proves (6.60).

Proof of (6.61). By Assumption M, {ht} and {ut} are mutually independent sequences.

Then

E(h2t − h2t )2 = E(K−1t

T∑j=1

b|t−j|h2j(u

2j − 1)

)237

= K−2t

T∑j,k=1

b|t−j|b|t−k|E[h2jh2k]cov(u2j , u

2k)].

By Assumption H, E[h2jh2k] ≤ (E[h4j ][h

4k])

1/2 ≤ maxj E[h4j ] < ∞, and by definition of bj,

maxj |b|j|| ≤ C. Using stationarity of {u2j}, (6.52) and (6.41), we obtain

E[(h2t − h2t )2] = CK−2t

T∑j,k=1

b|t−j||cov(u2j , u2k)|

≤ CK−1t (K−1t

T∑j=1

b|t−j|)∞∑

k=−∞

|cov(u20, u2k)| ≤ CK−1t ≤ CH−1

which proves (6.61).

Proof of (6.56). Using (6.59), write

T−1∑T

t=k+1 h−2t (h2t − h2t )ζt

= T−1∑T

t=k+1 h−2t (h2t − h2t )ζt+T−1

∑Tt=k+1 h

−2t (h2t − h2t )ζt

= Q1,T +Q2,T .

We will show that

E|Q`,T | = O((H/T )2γ +H−1

), ` = 1, 2 (6.63)

which proves (6.56)

Notice that the sequence {h−2t (h2t − h2t )} is independent of {ζt} while the i.i.d. property

of {ut} implies Eζtζs = 0 for t 6= s. Therefore,

EQ21,T = T−2

T∑t,s=k+1

E[h−2t (h2t − h2t )h−2s (h2s − h2s)]E[ζtζs]

= T−2T∑

t=k+1

E[h−4t (h2t − h2t )2]Eζ2t = E[ζ21 ]T−2T∑

t=k+1

E[h−4t (h2t − h2t )2].

By Assumption M, Eζ21 = Eu41E(u21−k − 1)2 < ∞, and by Assumption H, mint h2t ≥ a > 0

a.s. Hence,

E[h−4t (h2t − h2t ))2] ≤ a−4E(h2t − h2t )2 ≤ C(H/T )2γ (6.64)

by (6.60). Hence

E|Q1,T | ≤ (EQ21,T )1/2 ≤ C(H/T )γT−1/2 ≤ 2C{(H/T )2γ + T−1} (6.65)

which proves (6.63) for Q1,T .

38

Next we prove (6.63) for Q2,T . Write

h2t − h2t = K−1t∑T

j=1 b|t−j|h2j(u

2j − 1),

Q2,T = T−1T∑

t=k+1

h−2t (h2t − h2t )ζt = T−1K−1t

T∑t=k+1

T∑j=1

b|t−j|h−2t ζth

2j(u

2j − 1)

= T−1K−1t

T∑t=k+1

T∑j=1: j>t+3k

[...] + T−1K−1t

T∑t=k+1

T∑j=1: j<t−3k

[...] + T−1K−1t

T∑t=k+1

T∑j=1: |j−t|≤3k

[...]

= R1,T +R2,T +R3,T .

We will show

E|R`,T | = O(H−1), ` = 1, 2, 3 (6.66)

which implies E|Q2,T | ≤ CH−1 which proves (6.63) for Q2,T .

To evaluate ER21,T , recall that by assumption {ht} and {ut} are mutually independent.

Then, since ut are i.i.d. random variables, for any j > t+ 3k, j′ > t′ + 3k,

E[{h−2t ζth2j(u

2j − 1)}{h−2t′ ζt′h

2j′(u

2j′ − 1)}]

= E[h−2t h2jh−2t′ h

2j′ ]E[ζt′ζt]E[(u2j − 1)(u2j′ − 1)] = 0 if t 6= t′ or j 6= j′;

= E[h−4t h4j ]E[ζ2t ]E[(u2j − 1)2] if t = t′, j = j′.

Hence,

ER21,T = T−2K−2t

T∑t=k+1

T∑j=1

b2|t−j|E[h−4t h4j ]E[ζ2t ]E[(u2j − 1)2]

≤ E[ζ21 ]E[(u21 − 1)2](maxt,j

E[h−4t h4j ])T−2K−2t

T∑t=k+1

T∑j=1

b2|t−j|.

Under Assumption M, E[ζ21 ] = E[u41(u21−k− 1)2] = E[u41]E[(u21−k− 1)2] <∞, under Assump-

tion H, h−4t ≤ a−4 <∞ and maxj E[h4j ] <∞, and maxj |b|j|| <∞. Thus,

ER21,T ≤ CT−2K−1t

T∑t=k+1

(K−1t

T∑j=1

b|t−j|) ≤ CT−1K−1t ≤ CT−1H−1 ≤ CH−2

in view of (6.41). Then E|R1,T | ≤ (ER21,T )−1/2 ≤ CH−1 which proves (6.66). The proof of

(6.66) for R2,T is similar as the proof for R1,T .

Finally, using similar arguments as above, we obtain

E|R3,T | ≤ T−1K−1t

T∑t=k+1

T∑j=1:|j−t|≤3k

b|t−j|E[h−2t h2j ]E|ζt(u2t − 1)|

39

≤ (maxt,k

E[h−2t h2j ])E|ζ1(u21 − 1)|(maxjb|j|)K

−1t (6k + 1) ≤ CH−1

which proves (6.66) for R3,T . This completes the proof of (6.56).

Proof of (6.57). Using (6.59), we can bound

E[(h2t − h2t )2|ζt|] ≤ 2E[(h2t − h2t )2|ζt|] + 2E[(h2t − h2t )2|ζt|]. (6.67)

From (6.60) and independence of {ht} and {ut} it follows

E[(h2t − h2t )2|ζt|] = E[|ζ1|]E[(h2t − h2t )2] ≤ C(H/T )2γ

where C does not depend on t,H, T . In addition, we will show that

E[(h2t − h2t )2|ζt|] ≤ CH−1 (6.68)

which together with (6.67 ) proves (6.57).

It remains to prove (6.68). Write

h2t − h2t = K−1t

T∑j=1

b|t−j|h2j(u

2j − 1) = vt +K−1t

(b0h

2t (u

2t − 1) + bkh

2t−k(u

2t−k − 1)

),

where vt = K−1t∑T

j=1:j 6=t,t−k b|t−j|h2j(u

2j − 1). Then,

(h2t − h2t )2 ≤ 2v2t + 4K−2t(b20h

4t (u

2t − 1)2 + b2kh

4t−k(u

2t−k − 1)2

).

By assumption {ht} and {ut} are mutually independent, and by construction, vt is indepen-

dent of ζt. Hence,

E[(h2t − h2t )2|ζt|] ≤ 2E[v2t ]E[|ζt|]+4K−2t

(b20E[h4t ]E[(u2t − 1)2|ζt|] + b2kE[h4t−k]E[(u2t−k − 1)2|ζt|]

).

Using similar argument as in the proof of (6.61) it follows that

Ev2t ≤ CH−1

where C does not depend on t,H, T , and (6.41) implies K−1t ≤ CH−1. Under Assumption M,

Eu6t < ∞ which together with the i.i.d.property of variables ut implies E|ζt| = E|ζ1| < ∞,

E[(u2t − 1)2|ζt|] = E[(u21 − 1)2|ζ1|] <∞, E[(u2t−k − 1)2|ζt|] = E[(u21−k − 1)2|ζ1|] <∞. Hence,

E[(h2t − h2t )2|ζt|] ≤ CH−1

which proves (6.68). This completes the proof of (6.57).

Proof of (6.58). This bound can be verified using the same arguments as in the proof of

(6.57). 2

40